The shapes of things

The Shapes of Things Advances in Design and Control SIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines Editor-in-Chief Ralph C Smith, North Carolina State University Editorial Board Athanasios C Antoulas, Rice University Siva Banda, Air Force Research Laboratory John Betts, The Boeing Company (retired) Stephen L Campbell, North Carolina State University Michel C Delfour, University of Montreal Fariba Fahroo, Air Force Office of Scientific Research J William Helton, University of California, San Diego Arthur J Krener, University of California, Davis Kirsten Morris, University of Waterloo John Singler, Missouri University of Science and Technology Series Volumes Walker, Shawn W., The Shapes of Things: A Practical Guide to Differential Geometry and the Shape Derivative Michiels, Wim and Niculescu, Silviu-Iulian, Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue- Based Approach, Second Edition Narang-Siddarth, Anshu and Valasek, John, Nonlinear Time Scale Systems in Standard and Nonstandard Forms: Analysis and Control Bekiaris-Liberis, Nikolaos and Krstic, Miroslav, Nonlinear Control Under Nonconstant Delays Osmolovskii, Nikolai P and Maurer, Helmut, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control Biegler, Lorenz T., Campbell, Stephen L., and Mehrmann, Volker, eds., Control and Optimization with Differential-Algebraic Constraints Delfour, M C and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition Hovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation Speyer, Jason L and Jacobson, David H., Primer on Optimal Control Theory Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical Approaches Speyer, Jason L and Chung, Walter H., Stochastic Processes, Estimation, and Control Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLAB Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach ¸ Adaptive Control Tutorial Ioannou, Petros and Fidan, Barıs, Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems Robinett III, Rush D., Wilson, David G., Eisler, G Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical Systems Huang, J., Nonlinear Output Regulation: Theory and Applications Haslinger, J and Mäkinen, R A E., Introduction to Shape Optimization: Theory, Approximation, and Computation Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems Gunzburger, Max D., Perspectives in Flow Control and Optimization Delfour, M C and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J William and James, Matthew R., Extending H∞ Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives The Shapes of Things A Practical Guide to Differential Geometry and the Shape Derivative Shawn W Walker Louisiana State University Baton Rouge, Louisiana Society for Industrial and Applied Mathematics Philadelphia Copyright © 2015 by the Society for Industrial and Applied Mathematics 10 All rights reserved Printed in the United States of America No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA Trademarked names may be used in this book without the inclusion of a trademark symbol These names are used in an editorial context only; no infringement of trademark is intended MATLAB is a registered trademark of The MathWorks, Inc For MATLAB product information, please contact The MathWorks, Inc., Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, info@mathworks.com, www.mathworks.com Publisher Acquisitions Editor Developmental Editor Managing Editor Production Editor Copy Editor Production Manager Production Coordinator Compositor Graphic Designer David Marshall Elizabeth Greenspan Gina Rinelli Kelly Thomas Lisa Briggeman Nicola Cutts Donna Witzleben Cally Shrader Techsetters, Inc Lois Sellers Library of Congress Cataloging-in-Publication Data Walker, Shawn W., 1976The shapes of things : a practical guide to differential geometry and the shape derivative / Shawn W Walker, Louisiana State University, Baton Rouge, Louisiana pages cm (Advances in design and control ; 28) Includes bibliographical references and index ISBN 978-1-611973-95-2 Symmetry (Mathematics) Geometry Shapes Surfaces (Technology) Geometry, Differential I Title QA174.7.S96W35 2015 516.3’6 dc23 2015010487 is a registered trademark I dedicate this to Mari and Jane W Contents Preface ix Introduction 1.1 Differential Equations on Surfaces 1.2 Differentiating with Respect to Shape 1.3 Abstract vs Concrete Presentation 1.4 Outline 1.5 Prerequisites 1.6 Notation 1 7 Surfaces and Differential Geometry 2.1 Preliminaries 2.2 The Parametric Approach 2.3 Regular Surface 2.4 The Tangent Space 2.5 Minimal Regularity? 9 14 18 27 31 The Fundamental Forms of Differential Geometry 3.1 First Fundamental Form 3.2 Second Fundamental Form 3.3 Conclusion 33 33 43 56 Calculus on Surfaces 4.1 Functions on Surfaces 4.2 Differential Operators on Surfaces 4.3 Other Curvature Formulas 4.4 Integration by Parts 4.5 Other Identities 4.6 PDEs on Surfaces 59 59 61 71 74 77 82 Shape Differential Calculus 5.1 Introduction 5.2 General Framework 5.3 Derivatives of Functions with Respect to Flow Map Φε 5.4 Derivatives of Functions with Respect to Flow Map Xε 5.5 Basic Identities 5.6 Shape Perturbation of Functionals 87 87 87 90 91 93 101 vii viii Contents Applications 6.1 Minimal Surfaces 6.2 Surface Tension 6.3 Gradient Flows 6.4 Mean Curvature Flow 6.5 Image Segmentation 6.6 Conclusion 105 105 107 113 115 117 122 Willmore Flow 7.1 The Energy 7.2 Perturbation Analysis 7.3 Gradient Flow 123 123 124 133 A Vectors and Matrices 137 A.1 Vector Operations 137 A.2 Matrix Operations 137 A.3 Vector and Matrix Identities 140 B Derivatives and Integrals 141 B.1 Differential Formulas 141 B.2 Integral Formulas 142 Bibliography 145 Index 153 Preface Form follows function This old adage from art and architecture, credited to American architect Louis H Sullivan, holds true The shape of an object is intimately connected to its purpose Nature provides many examples of this: the shape of a tree and its leaves to harvest light, the wings of a bird to fly, the body of a snake to slither, and the structure of the human heart to keep us alive So good is this rubric that it finds application in modern design principles, e.g., the shapes of tools, the profile of an automobile, and the design of a bridge In an 1896 essay, Sullivan wrote form ever follows function and this is the law Sullivan means that form depends completely on function But what about the reverse? If an object’s shape changes, how is its function affected? Is the object’s function improved? Is the object better? In other words, does it make sense to consider function as dependent on shape? In a certain context, yes The main purpose of this book is to explain how to differentiate a function (in the calculus sense) with respect to a “shape variable.” This book is written to be as self-contained as possible It can be read by undergraduates who have completed the usual introductory calculus-based math courses It can be read by experts from other fields who wish to learn the fundamentals of differential geometry and shape differential calculus and apply them in their own disciplines It also makes a useful reference text for a variety of shape differentiation formulas Chapter gives more details on the prerequisites, framework, and overall philosophy of the book This book started as a set of notes I had created for my own use Over time, I continued to refine them and used them in a special topics course I taught at Louisiana State University (LSU) in Fall 2011 Eventually, after sharing the notes I realized their potential value to others and sought to create this book to make shape derivatives accessible to a broader audience Acknowledgments I would like to thank the following people for reading earlier versions of the text and making useful comments: Harbir Antil, Christopher B Davis, and Antoine Laurain I especially want to thank Mari Walker for proofreading the entire book I thank the anonymous reviewers; their compliments and criticisms certainly improved the manuscript I gratefully acknowledge prior support by NSF grants DMS1115636 and DMS-1418994 ix Chapter Introduction 1.1 Differential Equations on Surfaces The purpose of this book is to present an overview of differential geometry, which is useful for understanding mathematical models that contain geometric partial differential equations (PDEs), such as the surface (or manifold) version of the standard Laplace equation In particular, this requires the development of the so-called surface gradient and surface Laplacian operators These are nothing more than the usual gradient ∇ and Laplacian Δ = ∇ · ∇ operators, except they are defined on a surface (manifold) instead of standard Euclidean space (i.e., n ) One advantage of this approach is that it provides alternative formulas for geometric quantities, such as the summed (mean) curvature, that are much clearer than the usual presentation of texts on differential geometry 1.2 Differentiating with Respect to Shape The approach to differential geometry in this book is advantageous for developing the framework of shape differential calculus, which is the study of how quantities change with respect to changes of an independent “shape variable.” 1.2.1 A Simple Example The following example requires only the tools of freshman calculus Let f = f (r, θ) be a smooth function defined on the disk of radius R in terms of polar coordinates Denote the disk by Ω and let be the integral of f over Ω, i.e., R 2π = Ω f = f (r, θ) r d r d θ (1.1) Clearly, depends on R Let us assume f also depends on R, i.e., f = f (r, θ; R) A physical example could be that is the net flow rate of liquid through a pipe with crosssection Ω In this case, f is the flow rate per unit area and could be the solution of a PDE defined on Ω, e.g., a Navier–Stokes fluid flowing in a circular pipe It can be advantageous to know the sensitivity of with respect to R, e.g., for optimization purposes In other words, if R increases, how does change? To see this, let us Chapter Introduction differentiate with respect to R: d dR 2π = 2π R d dR f (r, θ; R) r d r d θ R = 2π f (r, θ; R) r d r d θ + 0 f (R, θ; R) R d θ, where f is the derivative with respect to R The dependence of f on R can more generally be viewed as dependence on Ω, i.e., f (·; R) ≡ f (·; Ω) Rewriting the above formula using Cartesian coordinates x, we get d dR = Ω f (x; Ω) d x + ∂Ω f (x; Ω) d S(x), (1.2) where d x is the “volume” measure and d S(x) is the “surface area” measure 1.2.2 More General Perturbations Let ν be the unit normal vector of ∂ Ω (pointing outward) We can view increasing R as a velocity field V that drives points on ∂ Ω in the normal direction, i.e., take V = ν on ∂ Ω Hence, (1.2) becomes d dR = Ω f (x; Ω) d x + ∂Ω f (x; Ω) V(x) · ν(x) d S(x), (1.3) where we view V as a velocity field that instantaneously perturbs the domain Ω We often call V a domain perturbation Let us adopt the notation f (x; Ω) ≡ f (Ω) and f (x; Ω) ≡ f (Ω; V), where f is called the shape derivative of f with respect to the domain perturd bation V Similarly, let us use δ (Ω; V) ≡ d R to denote the shape perturbation of with respect to Ω, in the direction V (i.e., a directional derivative) Thus, we obtain δ (Ω; V) = Ω f (Ω; V) + ∂Ω f (Ω)(V · ν), (1.4) which is formula (5.47) in Chapter Hence, we have derived (5.47) for the case where Ω is a disk perturbed by a velocity field V that causes Ω to uniformly expand (in the normal direction) The main purpose of this book is to derive (1.4), and other similar formulas, for general domains Ω and general choices of the perturbation V The framework of shape differential calculus provides the tools for developing the equations of mean curvature flow and Willmore flow, which are geometric flows that occur in many applications such as fluid dynamics and biology See Chapters and for examples 1.2.3 Sequential Optimization of Shape Which Way Is Down? It is obvious how to go down a hill As long as you can see and feel the ground, it is clear which direction to move in order to lower your elevation As motivation for the next section, let us view this as an optimization task In other words, let f = f (x, y) be a function describing the surface height of the hill, where (x, y) are the coordinates of our 140 Appendix A Vectors and Matrices Let en be the nth unit basis vector in for ≤ n ≤ Using (A.5) and (A.15), we have en · [(Ma) × (Mb)] = [en × (Ma)] · (Mb) = bT MT [en ]× M a (A.20) skew symmetric From (A.19), one can show that MT [en ]× M 3 ij = k=1 l =1 mki [en ]× kl m l j = [CT en ]× ij (A.21) Thus, we get en · [(Ma) × (Mb)] = b · [CT en ]× a = ((CT en ) × a) · b = (CT en ) · (a × b) = en · C(a × b) for ≤ n ≤ 3, (A.22) which is the assertion A.3 Vector and Matrix Identities n Proposition 31 Let a, b, c, d be column vectors in Then, , and let A, B, C, D be n ×n matrices (a ⊗ b) : (c ⊗ d) = (a · c)(b · d), (AB) : C = (AT C) : B, T (A.23) T (a C) · (b D) = D : [(b ⊗ a)C] Proof The first identity follows by (A.11) The second identity follows by (AB) : C = k ci j bk j = (AT C) : B k b k j c i j = i j k k j i For the last identity, we have (aT C) · (bT D) = (ak ck j ) j = (bi ak ck j ) = D : [(b ⊗ a)C] di j i j (bi di j ) i k k Appendix B Derivatives and Integrals This appendix gives some basic facts involving derivatives and integrals B.1 Differential Formulas B.1.1 Basic Identities Proposition 32 (derivative of an inverse matrix) Let G(t ) be an invertible matrix for all t Moreover, assume G and G−1 are continuously differentiable with respect to t Then, d d −1 G (t ) = −G−1 (t ) G(t ) G−1 (t ) dt dt (B.1) Proof It follows by 0= d d −1 d G−1 (t )G(t ) = G (t ) G(t ) + G−1 (t ) G(t ) dt dt dt Lemma B.1 (derivative of a determinant) Let G(t ) be a family of matrices that depend on the parameter t Assume that det(G(t )) = for all t Then, d det(G(t )) = det(G(t )) · trace dt d G(t ) G−1 (t ) dt (B.2) See the standard references for a proof Note that G−1 is the matrix inverse Lemma B.2 (expansion of the determinant) Consider det G(t ), where G(t ) = I + t A + t B + F(t ) and A, B are fixed matrices Suppose F(t ) = o(t ) Then, det G(t ) = + t trace A + t trace B + (trace A)2 − trace(A2 ) + o(t ) (B.3) Proof Let g (t ) = det G(t ) and a Taylor expansion From (B.2), we get d G(t ) G−1 (t ) , dt = g (t )trace A + t B + F (t ) G−1 (t ) , g (t ) = det(G(t ))trace ⇒ g (0) = trace A 141 (B.4) 142 Appendix B Derivatives and Integrals and g (t ) = g (t )trace A + t B + F (t ) G−1 (t ) + g (t )trace B + F (t ) G−1 (t ) + A + t B + F (t ) (G−1 (t )) (B.1) → = g (t )trace A + t B + F (t ) G−1 (t ) + g (t )trace B + F (t ) G−1 (t ) −1 (B.5) −1 − g (t )trace A + t B + F (t ) G (t )G (t )G (t ) g (0) = (trace A)2 + trace B − trace(A2 ) ⇒ So the second order Taylor expansion of g (t ) gives g (t ) = g (0) + g (0)t + g (0)t + o(t ) 2 = + t trace A + t trace B + (trace A)2 − trace(A2 ) + o(t ) (B.6) B.1.2 Change of Variables for Spatial Gradients Mapping a Domain Let Ω and Ω be open sets of points contained in n ; we call them domains Let Φ : Ω → Ω be a homeomorphism (see section 2.1.4) that maps points in the domain Ω to the domain Ω Thus, given a in Ω, there is a distinct point y in Ω such that y = Φ(a) and Φ−1 (y) = a Therefore, Ω = Φ(Ω), i.e., Φ maps the entire domain Ω onto Ω Similarly, Φ−1 maps the entire domain Ω onto Ω In addition, assume that Φ (and its inverse Φ−1 ) are at least C (differentiable) The Formula Let fˆ be a scalar function defined on Ω, i.e., fˆ ≡ fˆ(a) for all a in Ω With this, define f on Ω by f (y) = fˆ(Φ−1 (y)) = fˆ(a) Then, we have that gradients transform as ∂y j f (y) = ∂y j fˆ(Φ−1 (y)) = ⇒ i ∂ai fˆ(a) ∂ (Φ−1 (y) · ei ) a=Φ−1 (y) ∂y j (∇y f )(y) = ∇a fˆ(Φ−1 (y)) (∇y Φ−1 )(y), row vector row vector (B.7) matrix which implies that (∇y f ) ◦ Φ(a) = ∇a fˆ(a)(∇y Φ−1 ◦ Φ)(a) using (5.24) → = ∇a fˆ(a) [∇a Φ(a)]−1 row vector (B.8) matrix B.2 Integral Formulas B.2.1 Change 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Δt , 115 trace, 138 T ,7 Tb (·), 92 V, 2, 90 , 113 0, 137 active contours, 119 affine map, 13 allowable coordinate transformation, 18 arc-length, 33 area functional, 105, 116 atlas, 21 ball, 10 bending energy Euler–Lagrange equation, 129 functional, 123 general perturbations, 132 gradient flow, 134 normal perturbations, 129 volume and area constraints, 133 bijective, 12 boundary of a set, 10 boundary, surface without, 22 Chan–Vese functional, 118 153 change of parameters, 27 change of variables for gradient, 142 for integrals, 142 chart, 19 closed surface, 23 compact, 21 definition, 11 support, 11 compactly contained, 11 constant curvature surface, 108 constant normal extension, 92 continuous, 13 coordinate curves, 29 curvature Gauss, 52, 73 mean, 52 summed, 52, 71 vector, 52, 71 curvature (1-D curve), 48 curvature tensor, 45 curvature tensor transformation, 47 derivative material, 90, 92 of determinant, 141 of inverse matrix, 141 shape, 2, 91, 93 determinant, expansion of the, 141 diffeomorphism, 88 differentiable, 59 differentiable map, 88 domain regularity, 87 drag minimization, droplet equilibrium, 111 Euclidean space, extension, 92 extrinsic vs intrinsic, 154 Index first order optimality conditions, 108 fundamental form first, 34 second, 44, 47 matrices, 137 mean curvature flow, 116 metric tensor, 36 metric tensor transformation, 38 minimal surface, 106 gradient Euclidean, of inverse map, 93 shape, 88 surface, 62 gradient flow velocity, 113 gradient flow velocity space, 114 gradient flow, time-discrete, 115 normal curvature, 48 normal vector, 38 notation, hold-all domain, 89 homeomorphism, 13, 19 image features, 118 image point, 12 image segmentation, 117 immersion, 16 injective, 12 integral notation, integration by parts, 74 inverse image point, 12 isoperimetric inequality, 109 Jacobian, 16, 18 Lagrangian shape functional, 108 Laplace–Beltrami operator, 67 level set, 23 mapping, 12 inverse, 12 material derivative definition, 90, 92 normal vector, 96 summed curvature squared, 123 summed curvature vector, 97 surface Jacobian, 96 material point, 13 orientable surface, 40 orthogonal coordinates, 42 orthogonal transformation, 13 parameterization, allowable, 16 parametric representation, 15 surface, 17 perturbation domain, 2, 89 identity, 90 shape, 2, 101 prescribed mean curvature equation, 107 principal curvatures and directions, 50 red blood cell, 134 reference domain, 15 regular point, 17 regular surface, 19 Riemannian space, 36 rigid motion, 13 sets (basics), 10 shape derivative definition, 91, 93 normal vector, 99 position vector, 98 summed curvature, 100 shape descent direction, 114 shape functional, 88, 101 perturbation, 101 perturbation structure theorem, 104 shape operator, 51, 78 shape optimization, 3, 113 shape perturbation surface measure, 98 shape sensitivity, 101 soap film, 105 substrate surface tension, 109 support, 11 surface, 14 surface area, 42 surface divergence, 66 of normal vector, 80 surface gradient, 62 of normal vector, 78 surface heat equation, 83 surface Laplacian, 67 of normal vector, 80 surface partial differential equations (PDEs), 82 surface tension, 107 surface types: elliptic, hyperbolic, parabolic, planar, 53 surface without boundary, 22 surjective, 12 tangent plane, 17, 30 tangent plane/space, 28 tangent space projection, 65 tangent vector, 29 tangential directional derivative, 61 divergence, 66 gradient, 62 Laplacian, 67 vector field, 61 transformation, 12 tubular neighborhood, 92 vectors, 137 volume constraint, 117 Willmore functional, 123 ... and exterior of S (c) The closure of the set is shown, which is the union of the open set and its boundary: S¯ = S ∪ ∂ S Another example is the closed set [0, 1] ⊂ , i.e., the set of numbers between... body of a snake to slither, and the structure of the human heart to keep us alive So good is this rubric that it finds application in modern design principles, e.g., the shapes of tools, the profile... other are said to be homeomorphic Sets that are homeomorphic have the “same topology”, i.e., their connectedness is the same; they have the same kinds of “holes.” There is further discussion of